Shelf Load & Sag Calculator
Calculate the maximum mid-span deflection (sag) and bending stress of a shelf treated as a simply supported beam under a uniform distributed load.
mm
mm
mm
kg
Formulas Used
Second Moment of Area (rectangular section):
I = b · h³ / 12
UDL intensity:
w = W / L (W in Newtons, L in mm)
Maximum deflection – simply supported beam, UDL (mid-span):
δmax = 5 · w · L⁴ / (384 · E · I)
Maximum bending moment (mid-span):
Mmax = w · L² / 8
Maximum bending stress (extreme fibre):
σmax = Mmax · (h/2) / I
Maximum shear force (at supports):
Vmax = W / 2
Assumptions & References
- The shelf is modelled as a simply supported beam (pinned at both ends) – conservative for shelves with brackets.
- The load is assumed to be uniformly distributed along the full span.
- The cross-section is rectangular (solid); hollow or composite sections require a different I.
- Material behaviour is assumed linear-elastic; no plastic deformation or creep is modelled.
- Serviceability limits follow common practice: δ ≤ L/360 (preferred) and δ ≤ L/200 (maximum acceptable) per general structural guidelines.
- Self-weight of the shelf is not included in the load W; add it manually if significant.
- Reference: Timoshenko & Gere, Mechanics of Materials; AS 1170.1 for load combinations.
- Typical E values: Steel 200,000 MPa; Aluminium 70,000 MPa; Structural Plywood 12,000 MPa; MDF/Particleboard 3,500 MPa.